Whakaoti i te 5v^2+45v+70

$$5 v ^ { 2 } + 45 v + 70$$

Tauwehe

$$5\left(v+2\right)\left(v+7\right)$$

Ngā Upane Otinga

Tauwehea te 5.

$$5\left(v^{2}+9v+14\right)$$

Whakaarohia te v^{2}+9v+14. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei v^{2}+av+bv+14. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=9 ab=1\times 14=14$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 14.

$$1,14 2,7$$

Tātaihia te tapeke mō ia takirua.

$$1+14=15 2+7=9$$

Ko te otinga te takirua ka hoatu i te tapeke 9.

$$a=2 b=7$$

Tuhia anō te v^{2}+9v+14 hei \left(v^{2}+2v\right)+\left(7v+14\right).

$$\left(v^{2}+2v\right)+\left(7v+14\right)$$

Tauwehea te v i te tuatahi me te 7 i te rōpū tuarua.

$$v\left(v+2\right)+7\left(v+2\right)$$

Whakatauwehea atu te kīanga pātahi v+2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(v+2\right)\left(v+7\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$5\left(v+2\right)\left(v+7\right)$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ka taea te huamaha pūrua te tauwehe mā te whakamahi i te huringa ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), ina ko x_{1} me x_{2} ngā otinga o te whārite pūrua ax^{2}+bx+c=0.

$$5v^{2}+45v+70=0$$

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$v=\frac{-45±\sqrt{45^{2}-4\times 5\times 70}}{2\times 5}$$

Pūrua 45.

$$v=\frac{-45±\sqrt{2025-4\times 5\times 70}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$v=\frac{-45±\sqrt{2025-20\times 70}}{2\times 5}$$

Whakareatia -20 ki te 70.

$$v=\frac{-45±\sqrt{2025-1400}}{2\times 5}$$

Tāpiri 2025 ki te -1400.

$$v=\frac{-45±\sqrt{625}}{2\times 5}$$

Tuhia te pūtakerua o te 625.

$$v=\frac{-45±25}{2\times 5}$$

Whakareatia 2 ki te 5.

$$v=\frac{-45±25}{10}$$

Nā, me whakaoti te whārite v=\frac{-45±25}{10} ina he tāpiri te ±. Tāpiri -45 ki te 25.

$$v=-\frac{20}{10}$$

Whakawehe -20 ki te 10.

$$v=-2$$

Nā, me whakaoti te whārite v=\frac{-45±25}{10} ina he tango te ±. Tango 25 mai i -45.

$$v=-\frac{70}{10}$$

Whakawehe -70 ki te 10.

$$v=-7$$

Tauwehea te kīanga taketake mā te whakamahi i te ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Me whakakapi te -2 mō te x_{1} me te -7 mō te x_{2}.

$$5v^{2}+45v+70=5\left(v-\left(-2\right)\right)\left(v-\left(-7\right)\right)$$

Whakamāmātia ngā kīanga katoa o te āhua p-\left(-q\right) ki te p+q.

$$5v^{2}+45v+70=5\left(v+2\right)\left(v+7\right)$$

Aromātai

$$5\left(v+2\right)\left(v+7\right)$$